
It is not hard to see that the image of \(\overline{{\cal M}}_{\mathbf {r}}\)  under \(\phi \)  is \({\cal M}_{\mathbf {r}}\) .
Let \(\bar{{\cal X}} = \left\lbrace {\cal X}^1,\cdots ,{\cal X}^d\right\rbrace \in \overline{{\cal M}}_{\mathbf {r}}\)  and \(\bar{{\cal Y}} = \left\lbrace {\cal Y}^1,\cdots ,{\cal Y}^d\right\rbrace \in \overline{{\cal M}}_{\mathbf {r}}\) . Then by Proposition 7 in  [1]},
\(\phi \left(\bar{{\cal X}}\right)= \phi \left(\bar{{\cal Y}}\right)\)  if and only if there exist invertible matrices \(\mathbf {A}_1,\cdots ,\mathbf {A}_{d-1}\)  of appropriate sizes such that
\(\bar{{\cal Y}} = \left\lbrace {\cal X}^1\times _3\mathbf {A}_1^\mathsf {T},{\cal X}^2\times _1\mathbf {A}_1^{-1}\times _3\mathbf {A}_2^\mathsf {T},\cdots ,{\cal X}^d\times _1\mathbf {A}_{d-1}^{-1}\right\rbrace .\) 
